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Fatigue damage in solder joint interconnects - presentation

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  • 1. Fatigue damage modeling in solder interconnects using a cohesive zone approach Adnan Abdul-Baqi, Piet Schreurs, Marc Geers AIO-Meeting: 03-06-2003 Supported by Philips
  • 2. Outline• Introduction• Geometry and loading• Cohesive zone method: – Cohesive zone formulation – Cohesive tractions – Damage evolution law – One dimensional example• Results: – Damage distribution – Corresponding total effective damage and reaction force – Life-time prediction in comparison with empirical models• Conclusions
  • 3. Printed circuit board (PCB)• Solder joints provide mechanical & electrical connection between the silicon chip and the printed circuit board.• Repeated switching of the device → temperature fluctuations → fatigue of the solder joints → device failure.
  • 4. Solder bump• Interconnects failure contributes by up to 20 % to device failure.
  • 5. Tin-Lead solder Typical Tin-Lead microstructure (A. Matin).• Simplified microstructure is chosen for the simulations: – Physically: rapid coarsening → continuous change. – Numerically: Large number of degrees of freedom → time consuming.
  • 6. Geometry and loading: solder bump Ux 0.1 mm Lead y Tin x 0.1 mm• Plane strain formulation, thickness = 1 mm.• Elastic properties: Tin (E = 50 GPa, ν = 0.36), Lead (E = 16 GPa, ν = 0.44) . max• Loading: cyclic mechanical with Ux = 1 µm.
  • 7. Cohesive zone method: cohesive zone? continuum element 3 n 4 t ∆ cohesive zone 1 2 continuum element• Cohesive zones are embedded between continuum elements.• Constitutive behavior: specified through a relation between the separation ∆ (initially = 0) and a corresponding traction T(∆).
  • 8. Cohesive zone method: stiffness matrix and nodal force vector• The cohesive zone nodal displacement vector is constructed in the local frame of reference (t,n): uT = {u1, u1 , u2, u2 , u3, u3 , u4, u4 }. t n t n t n t n• The relative displacement vector ∆ is then calculated as:     ∆  t  ∆= = Au  ∆n   where A is a matrix of the shape functions:   −h1 0 −h2 0 h1 0 h2 0 A=  0 −h1 0 −h2 0 h1 0 h2   and 1 1 h1 = (1 − η), h2 = (1 + η). 2 2 The parameter η is defined at the cohesive zone mid plane and varies between −1 at nodes (1,3) and 1 at nodes (2,4).
  • 9. • The cohesive zone internal nodal force vector and stiffness matrix are now writ- ten as: l +1 f = S ATT dS = −1 ATT dη 2 l +1 K = S ATBA dS = −1 ATBA dη 2 where S is the cohesive zone area, l is the cohesive zone length and B is the cohesive zone constitutive tangent operator given by: ∂Tt ∂Tt     ∂∆t ∂∆n     B=  .  ∂Tn ∂Tn        ∂∆t ∂∆n• Finally, K and f are transformed to the global frame of reference (x,y).
  • 10. Cohesive tractions: monotonic loading 1 1Tn/σmax 0 Tt/τmax 0 −1 −2 (a) −1 (b) −1 0 1 2 3 4 5 6 −3 −2 −1 0 1 2 3 ∆ /δ ∆ /δ n n t t Cohesive zone monotonic normal (a) and shear (b) tractions.• Characteristics: peak traction and cohesive energy.• The softening branch is the energy dissipation source.
  • 11. Cohesive tractions: cyclic loading• A linear relation is assumed between the cohesive traction and the corresponding cohesive opening: Tα = kα (1 − Dα )∆α where kα is the initial stiffness and α is either the local normal (n) or tangential (t) direction in the cohesive zone plane.• Energy dissipation is accounted for by the damage variable D.• The damage variable is supplemented with an evolution law: ˙ ˙ D = f (∆, ∆, T, D, ...).
  • 12. Cyclic loading: damage evolution• Evolution law (motivated by Roe and Siegmund, 2003):   |Tα | Dα = cα |∆α | (1 − Dα + r)m  ˙ ˙  − σf   1 − Dα where cα , r, m are constants and σf is the cohesive zone endurance limit.• Satisfies main experimental observations on cyclic damage: – Damage increases with the number of cycles. – The larger the load, the larger the induced damage. – Damage is larger in the presence of mean stress/strain. – Load sequencing: cycling at a high stress level followed by a lower level (H–L) causes more damage than when the order is reversed (L–H). σf = 0 −→ linear damage accumulation (Miner’s law).
  • 13. Cohesive zone: k = 106 GPa/mm, c = 100 mm/N, σf = 150 MPa, r = 10−3, m = 3.Continuum: E = 30 GPa, ν = 0.25.Loading: axial sinusoidal displacement U with amplitude of 0.2 µm.Geometry: L = 20 µm, R = 10 µm. ¡ ¡ ¡ ¡ ¡ ¡ ¡  ¢¡¡¡¡¡¡¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡  ¢¡¡¡¡¡¡¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¤¡¢¡¢¤¡¢¤¡¢¤¡¢¤¡¢¤¡ ¢ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢  ¡ ¡ ¡ ¡ ¡ ¡ ¡ £ £ £ £ £ £ ¤¡¡¤¡¤¡¤¡¤¡¤¡ ¢¡¢¡¢¡¢¡¢¡¢¡¢¡ ¢  ¡ ¡ ¡ ¡ ¡ ¡ ¡ £¡¢¡¢£¡¢£¡¢£¡¢£¡¢£¡ ¡¡¡¡¡¡¡ ¢ ¢£¤¤£¤¡¤£¡¤¡¤¡¤¡¤¡¤¡ £¡¡£¡£¡£¡£¡£¡ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡ £¡£¡£¡£¡£¡£¡£¡ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤£¤¡¤¡¤¡¤¡¤¡¤¡¤£¤ £¡¡£¡£¡£¡£¡£¡ ¤¡¡¤¡¤¡¤¡¤¡¤¡ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡¤£¡£¡£¡£¡£¡£¡¤£ ¤¡¡¤¡¤¡¤¡¤¡¤¡ £¡£¡£¡£¡£¡£¡£¡£ ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ L ¤¡¤¡¤¡¤¡¤¡¤¡¤¡¤ £¡£¡£¡£¡£¡£¡£¡£ ¤¡£¤¡¤¡¤¡¤¡¤¡¤¡£¤ £¡¡£¡£¡£¡£¡£¡ ¤¡£¤¡¤¡¤¡¤¡¤¡¤¡£¤ ¡¥¡¥¡¥¡¥¡¥¡¥¡ ¥¦¡¦¡¦¡¦¡¦¡¦¡¦¡£¤ ¡¥¡¥¡¥¡¥¡¥¡¥¡ ¥¦¡¡¡¡¡¡¡£¤ £¡¡£¡£¡£¡£¡£¡ ¡¦£¤£¤¡¦¡¦¡¦¡¦¡¦¡ ∆ ¡¥¡¥¡¥¡¥¡¥¡¥¡¦¥¦ ¦¥¦¡¡¡¡¡¡¡¥ ¦¡¦¡¦¡¦¡¦¡¦¡¦¡¥¦ ¨ ¨ ¨ ¨ ¨ ¨ §¡¡§¡§¡§¡§¡§¡ ¨¡¡¨¡¨¡¨¡¨¡¨¡ §¡¡§¡§¡§¡§¡§¡ ¥¡¥¡¥¡¥¡¥¡¥¡¥¡ ¡¡¡¡¡¡¡¥¦¥¦¨§¨§¨ §¥¡¥¡¥¡¥¡¥¡¥¡¥¡ ¡¦¡¦§¡¦§¡¦§¡¦§¡¦§¡ ¨¡¡¨¡¨¡¨¡¨¡¨¡ ¡¦§¨¡¦¨¡¦¨¡¦¨¡¦¨¡¦¨¡ ¡¨¡§¡§¡§¡§¡§¡ §¨¡¨§¡¨¡¨¡¨¡¨¡¨¡ ¡¡§¡§¡§¡§¡§¡ §¨¡¨§¡¨¡¨¡¨¡¨¡¨¡¨§ ¡¨§¡§¡§¡§¡§¡§¡¨§ §§¨¡¡§¡§¡§¡§¡§¡ ¡§¡¨¡¨¡¨¡¨¡¨¡§ ¨¨¡¨¡¨¡¨¡¨¡¨¡¨¡¨ §¡¡§¡§¡§¡§¡§¡ ¡¨§¡§¡§¡§¡§¡§¡¨§ ¨¡¡¨¡¨¡¨¡¨¡¨¡ §¡¨§¡¨¡¨¡¨¡¨¡¨¡¨§ ¡§¡§¡§¡§¡§¡§¡§ §¨¡¡¨¡¨¡¨¡¨¡¨¡ ¡¨§¡§¡§¡§¡§¡§¡§¨ §¨¡¡¨¡¨¡¨¡¨¡¨¡ ¡¨§¨¡§¡§¡§¡§¡§¡¨§¨ ¨§¨¡¡¨¡¨¡¨¡¨¡¨¡ §¡§¡§¡§¡§¡§¡§¡§ ¨¡¨¡¨¡¨¡¨¡¨¡¨¡¨ §¡§¡§¡§¡§¡§¡§¡§ R ¨¡¨¡¨¡¨¡¨¡¨¡¨¡¨ §¡¡§¡§¡§¡§¡§¡ ¡§¨§¨§¡¡¡¡¡¡§¨§¨§ Uniaxial cyclic tension-compression example
  • 14. Initial cohesive stiffnessHigh initial stiffness → minimize artificial enhancement of the overall compliance.For a bar containing n equally spaced cohesive zones: (U − n∆) σ= E, L T = k(1 − D)∆.Stress continuity → σ = (U/L)E ∗, where E ∗ is given as:   1 E ∗ = 1 − kL   E.  nE (1 − D) + 1 nETo ensure a negligible enhancement of the overall compliance → kL << 1. EIn a two-dimensional model the condition is estimated by kl << 1, where l ≈ L/nis the average cohesive zone length.
  • 15. 400 0.15 (a) 0.1 (b) 200 T (MPa) 0.05F (N) 0 0 −0.05 −200 −0.1 −0.15 −400 0 200 400 600 800 1000 −0.05 0 0.05 0.1 0.15 0.2 N (cycles) ∆ (µ m) (a) Reaction force vs. cycles to failure. (b) Cohesive traction vs. opening. • Assumption: damage does not occur under compression: – Physically: infinite compressive strength. – Numerically: minimizes inter-penetration (overlapping) of neighboring con- tinuum elements under compression.
  • 16. F versus N : experimental (Erik de Kluizenaar: Philips).
  • 17. 0.15 0.15 0.1 (a) 0.1 (b) 0.05 0.05F (N) F (N) 0 0 −0.05 −0.05 −0.1 −0.1 −0.15 −0.15 0 20 40 60 80 100 0 500 1000 1500 2000 N (cycles) N (cycles) Different damage parameters: (a) r = 10−3, m = 1. (b) r = 0, m = 3.
  • 18. 1 1 H−L 0.8 0.8 L−H 0.6 εmean = 0 0.6 D εmean = 0.5 % D 0.4 0.4 0.2 (a) 0.2 (b) 0 0 0 200 400 600 800 1000 0 100 200 300 400 N (cycles) N (cycles) (a) Mean strain effect. (b) Load sequencing effect.H–L: 200 cycles at max = 1 % followd by 200 cycles at max = 0.5 %L–H: 200 cycles at max = 0.5 % followd by 200 cycles at max = 1 %
  • 19. Cohesive parameters: solder bump czg1 czg2 czg3 czg4• Initial cohesive zone stiffness kα = 106 GPa/mm. – Sufficiently high compared to continuum stiffness. Identical for all cohesive zone groups.• Damage coefficient cα in [mm/N]: czg1 : 0, czg2 : 25, czg3 : 100, czg4 : 0.• σα = 0 MPa, r = 10−3. f
  • 20. Computational time reduction• Loading is applied incrementally.• For large number of cycles → time consuming.• Computational time reduction: only selected cycles are simulated.• Time reduction of more than 90 % in some cases.
  • 21. Results: damage distribution N = 500; Deff = 0.14 N = 1000; Deff = 0.22Damage distribution in the solder bump at different cycles. Red lines indicate i damaged cohesive zones (Deff ≥ 0.5). 2 2 Deff = (Dn + Dt − DnDt )1/2 i i i i i
  • 22. N = 2000; Deff = 0.31 N = 8000; Deff = 0.4
  • 23. 0.5 0.4 0.3 eff D 0.2 0.1 0 0 2000 4000 6000 8000 N (cycles) The total effective damage versus the number of cycles.The total effective damage is calculated by averaging over all cohesive zones: 1 N Deff = Deff S i i S i i where Deff is the effective damage at cohesive zone (i).
  • 24. 8 6 4 F (N) 2 0 x −2 −4 −6 −8 0 2000 4000 6000 8000 N (cycles) The reaction force versus the number of cycles.• Slow softening followed by rapid softening (Kanchanomai et al., 2002)
  • 25. S-N curve −0.5 FEM ) −1 linear fit max −1.5 log(ε −2 −2.5 −3 1 2 3 4 5 6 log(2N ) fApplied strain max versus the number of reversals to failure 2Nf .
  • 26. • Finite element data can be fitted with the Coffin-Manson model: max = a(2Nf )b a: fatigue ductility coefficient b: fatigue ductility exponent• Failure criteria: 50% reduction in the reaction force −→ a = 0.83, b = −0.49.• Reduction of 25% or 75% → same value of b.• Change by ±50 % in the Young’s modulii → same value of b.
  • 27. Effect of the elastic parameters 4 3 r Nf/Nf 2 1 0 0.5 0.75 1 1.25 1.5 E/ErVariation of Nf with E at max = 1%. Fitting curve: Nf /Nfr = (E/E r)−1.83.
  • 28. Conclusions• Evolution law captures main cyclic damage characteristics.• The model’s prediction of the solder bump life-time agrees with the Coffin- Manson model.• More efficient computational time reduction scheme: −→ simulation of larger number of cycles. −→ more realistic microstructure.
  • 29. Movie ...
  • 30. Thank youQuestions?